One problem with the final act is that it's Spock who defeats Khan rather than Kirk. Kirk has been the primary character throughout the film; it is Kirk who goes on a journey from angry jingoist to realizing he's been had by Marcus... and then at the end, Kirk does not resolve the final conflict.
But my biggest problem with Into Darkness is actually the presence of Khan himself. Not the function Khan serves in the story -- seeming ally turned menace -- but his identity as Khan. Simply put, the Khan character is a legacy of 1960s era racism -- having a Mexican man of European heritage portraying a Sikh from the Indian subcontinent is... well, it's pretty damn problematic. I love Ricardo Montalban, you love Ricardo Montalban, we all love Ricardo Montalban, but his casting definitely played into white American stereotypes of the generic foreigner and (moreso in "Space Seed" than in TWOK) the sexually threatening brown man come to "take our women." We can certainly grade TOS on a curve and grandfather it in because things weren't as racially enlightened as they are today, but "problematic" is built into the character from the start.
The options for using the Khan character in 2013 are, well, not great. The role calls for Khan to essentially do an outer space version of 9/11 at the end of the film, crashing the USS Vengeance into downtown San Francisco. This presents us with a problem if the role is Khan: Sikhs worldwide have struggled with racists who stereotype them as terrorists (particularly confusing their turbans with stereotypes of Arabs). If Khan is played by someone with actual Sikh/Indian subcontinent heritage, it is, furthermore, generally problematic, because you've got a brown man doing terrorism in a major motion picture, and that just reinforces existing racist stereotypes.
On the other hand, the option they chose for using Khan is problematic in its own right: they literally whitewashed the role, casting a white Englishman to play a character canonically identified as Sikh Indian. This A) makes no sense unless some plot device is introduce to explain the difference, and B) is another example of white actors appropriating roles for persons of color.
The fundamental solution to the problem, of course, is to "kill your darling:" Recognize that the narrative does not actually require the "ally turned enemy created by blowback to imperialism" role to be Khan per se, and make it a different character without that kind of problematic baggage. Which STID did not do. Oh well.
